This dish is still prepared in Andalusia and bears the name of "gazpachuelo."
Heat up the salted water. Once it boils, pour in the egg white. Drain and set aside, keeping the hot water.
Make a mayonnaise-type sauce by beating the egg yolk in a large bowl with a whisk or fork.
Gradually add olive oil. Beat for 10-15 minutes until firm.
In another bowl, take one tablespoon of the water used to cook the egg white and add two tablespoons of cold water.
While mixing the mayonnaise, pour the water in very slowly. Add saffron and sugar (optional).
Gradually add the rest of the hot salted water to the mayonnaise. Be careful: if you pour too quickly, the egg yolk will coagulate.
Pour the gazpachuelo and pieces of cooked egg white into four bowls. Add fresh lemon juice and a lemon slice to each bowl.
Serve hot with croutons.